Some estimates for the Bergman Kernel and Metric in Terms of Logarithmic Capacity

Abstract

AbstractFor a bounded domain Ω on the plane we show the inequality cΩ(z)2 ≤ 2πKΩ(z), z ∈ Ω, where cΩ(z) is the logarithmic capacity of the complement ℂ\Ω with respect to z and KΩ is the Bergman kernel. We thus improve a constant in an estimate due to T. Ohsawa but fall short of the inequality cΩ(z)2 ≤ πKΩ(z) conjectured by N. Suita. The main tool we use is a comparison, due to B. Berndtsson, of the kernels for the weighted complex Laplacian and the Green function. We also show a similar estimate for the Bergman metric and analogous results in several variables.